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  "body": "\n## Summary\n\nA smooth map between [[smooth manifolds|F45VQ47GPV]] is stable if every map close enough to it is equivalent to it.\nHere equivalent means equal after a change of coordinates in the source and in the target.\nSmall perturbations do not change its singularities in kind, only move them a little.\nStable maps are the maps whose singularities survive in nature and in experiments, which is why [[singularity theory|42V5XGZS4F]] aims to classify them.\nMather proved that stability is equivalent to a linear condition, [[infinitesimal stability|C3TFCVE2YK]], for proper maps.\n\n## Explanations\n\n### L1 human\n\nImagine a map drawn with a slightly shaky hand.\nIf the shaky copy can always be turned back into the original by smoothly bending the source and the target, the map is stable.\nThe height function on a circle is stable: tilt or wobble the circle a little and it still has one highest and one lowest point, of the same kind.\nA constant function on a circle is not stable: a tiny wobble gives a function with peaks and valleys, which no change of coordinates can make constant again.\n",
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